Monica Set

The $n$th Monica set $M_n$ is defined as the set of Composite Numbers $x$ for which $n\vert S(x)-S_p(x)$, where

$$x=a_0+a_1(10^1)+\ldots+a_d(10^d)=p_1p_2\cdots p_n,$$ (1)

and

$$S(x) = \sum_{j=0}^d a_j$$ (2)

$$S_p(x) = \sum_{i=1}^m S(p_i).$$ (3)

Every Monica set has an infinite number of elements. The Monica set $M_n$ is a subset of the Suzanne Set $S_n$. If $x$ is a Smith Number, then it is a member of the Monica set $M_n$ for all $n\in\Bbb{N}$. For any Integer $k>1$, if $x$ is a $k$-Smith Number, then $x\in M_{k-1}$.

See also Suzanne Set

References

Smith, M. ``Cousins of Smith Numbers: Monica and Suzanne Sets.'' Fib. Quart. 34, 102-104, 1996.